Formation stresses can affect geophysical prospecting and development of oil and gas reservoirs. For example, overburden stress, maximum and minimum horizontal stresses, pore pressure, wellbore pressure and rock strength can be used to produce a failure model to aid in well planning, wellbore stability calculations and reservoir management. It is known that elastic wave velocities change as a function of prestress in a propagating medium. For example, sonic velocities in porous rocks change as a function of effective prestress. However, estimating formation stresses based on velocity can be problematic because of influences on the horizontal shear modulus C66. For example, the horizontal shear modulus C66 is reduced in the presence of horizontal fluid mobility in a porous reservoir. Generally, the tube wave slowness decreases by about 2 to 3% in the presence of fluid mobility, resulting in a decrease in C66 by about 4 to 6%. Conversely, the horizontal shear modulus C66 is increased in the presence of high clay content in a shale interval. Consequently, it is difficult to accurately estimate the ratio of vertical to horizontal stress ratios without compensating for the changes in C66.
Various devices are known for measuring formation characteristics based on sonic data. Mechanical disturbances are used to establish elastic waves in earth formations surrounding a borehole, and properties of the waves are measured to obtain information about the formations through which the waves have propagated. For example, compressional, shear and Stoneley wave information, such as velocity (or its reciprocal, slowness) in the formation and in the borehole can help in evaluation and production of hydrocarbon resources. One example of a sonic logging device is the Sonic Scanner® from Schlumberger. Another example is described in Pistre et al., “A modular wireline sonic tool for measurements of 3D (azimuthal, radial, and axial) formation acoustic properties, by Pistre, V., Kinoshita, T., Endo, T., Schilling, K., Pabon, J., Sinha, B., Plona, T., Ikegami, T., and Johnson, D.”, Proceedings of the 46th Annual Logging Symposium, Society of Professional Well Log Analysts, Paper P, 2005. Other tools are also known. These tools may provide compressional slowness, Δtc, shear slowness, Δts, and Stoneley slowness, Δtst, each as a function of depth, z, where slowness is the reciprocal of velocity and corresponds to the interval transit time typically measured by sonic logging tools. An acoustic source in a fluid-filled borehole generates headwaves as well as relatively stronger borehole-guided modes. A standard sonic measurement system uses a piezoelectric source and hydrophone receivers situated inside the fluid-filled borehole. The piezoelectric source is configured as either a monopole or a dipole source. The source bandwidth typically ranges from a 0.5 to 20 kHz. A monopole source primarily generates the lowest-order axisymmetric mode, also referred to as the Stoneley mode, together with compressional and shear headwaves. In contrast, a dipole source primarily excites the lowest-order flexural borehole mode together with compressional and shear headwaves. The headwaves are caused by the coupling of the transmitted acoustic energy to plane waves in the formation that propagate along the borehole axis. An incident compressional wave in the borehole fluid produces critically refracted compressional waves in the formation. Those refracted along the borehole surface are known as compressional headwaves. The critical incidence angle θi=sin−1(Vf/Vc), where Vf is the compressional wave speed in the borehole fluid; and Vc is the compressional wave speed in the formation. As the compressional headwave travels along the interface, it radiates energy back into the fluid that can be detected by hydrophone receivers placed in the fluid-filled borehole. In fast formations, the shear headwave can be similarly excited by a compressional wave at the critical incidence angle θi=sin−1(Vf/Vs), where Vs is the shear wave speed in the formation. It is also worth noting that headwaves are excited only when the wavelength of the incident wave is smaller than the borehole diameter so that the boundary can be effectively treated as a planar interface. In a homogeneous and isotropic model of fast formations, as above noted, compressional and shear headwaves can be generated by a monopole source placed in a fluid-filled borehole for determining the formation compressional and shear wave speeds. It is known that refracted shear headwaves cannot be detected in slow formations (where the shear wave velocity is less than the borehole-fluid compressional velocity) with receivers placed in the borehole fluid. In slow formations, formation shear velocities are obtained from the low-frequency asymptote of flexural dispersion. There are standard processing techniques for the estimation of formation shear velocities in either fast or slow formations from an array of recorded dipole waveforms. A different type of tool described in U.S. Pat. No. 6,611,761, issued Aug. 26, 2003 for “Sonic Well Logging for Radial Profiling” obtains radial profiles of fast and slow shear slownesses using the measured dipole dispersions in the two orthogonal directions that are characterized by the shear moduli c44 and c55 for a borehole parallel to the X3-axis in an orthorhombic formation.